Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation

Abstract

A formalism for systematically deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear PDEs is presented. Three algorithms are derived for the nonlinear Klein-Gordon equation (NLKGE) based on the proposed formalism. The local conservation laws of the NLKGE form the basic starting point in our derivation, which hinges essentially on the commutativity of certain finite difference operators. Such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the PDE level. With appropriate boundary conditions, the proposed algorithms preserve in the discrete sense either the total system energy or the system's linear momentum. Several variants of the present algorithms and their relation to previously proposed algorithms are discussed. An analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms. The preservation of energy of the present algorithms for the NLKGE can also be viewed as providing a method of stabilization for conditionally stable algorithms for the linear wave equation. The computer implementation of the proposed algorithms, with the treatment of the boundary conditions, is presented in detail. Numerical examples are given concerning soliton collisions in the sine-Gordon equation, the double sine-Gordon equation, and the φ ± 4 (‘phi-four’) equation. The numerical results demonstrate that the present algorithms can preserve accurately (up to 10 decimal digits) the total system energy for a very coarse grid. Reliable algorithms for Josephson junction models, which contain dissipation, damping mechanisms and driving bias current, are obtained as direct by-products of the proposed invariant-conserving algorithms for the NLKGE. Even though presented mainly for the 1-D case, the proposed algorithms are generalizable to the 2-D and 3-D cases, and to the case of complex-valued NLKGE.